Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
BrownianMotion, localtime
An estimator
based onquadraticvariation
Application tovolatility
modeling

Statistical estimation of the Oscillating
Brownian Motion and application to volatility
modeling
Paolo Pigato
Joint work with Antoine Lejay
INRIA Nancy, equipe TOSCA
Paris, 29/09/2016

Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling

1

Oscillating Brownian Motion, local time2

An estimator based on quadratic variation

3

Application to volatility modeling

Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

The oscillating Brownian Motion
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling

Consider the process in R solution ofZ t
Yt = Y0 +σ(Yt )dWt

Paolo Pigatowith

Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

0σ(x) =n σ+σ−for x ≥ 0for x < 0The process is defined using the recipe of Ito-McKean toconstruct a process with given speed measure and scale
function. It behaves like a Brownian motion which changes
variance parameter each time it crosses 0. In this talk we also
suppose Y0 = 0 a.s., and fix the final time horizon T = 1.
The aim of the present work is to propose and analyze someestimators for the parameters of such process.

Tanaka formula and local time
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtimeAn estimator
based onquadraticvariation
Application tovolatility
modeling

For any continuous semimartingale MZ t
|Mt | − |M0 | =sgn(Ms )dMs + LMt (0)0With x + = x ∨ 0; x − = (−x) ∨ 0, we also have
Z t1++(0)Mt − M0 =1(Ms ≥ 0)dMs + LM
2 t0
Z t
1
Mt− − M0− = −1(Ms < 0)dMs + LM(0)2 t0where1LMt (0) := lim
ε↓0 2εin the local time of M at 0.

Z t1(|Ms | ≤ ε)ds0

Statisticalestimation ofthe Oscillating
BrownianMotion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

We apply the formula for the positive part to the OBM Y :Z t
1+Yt =(0)1(Ys ≥ 0)dYs + LY
2 t0
Z t
1
= σ+(0)1(Ys ≥ 0)dWs + LY
2 t0
We hope to recover an estimator for σ+ from the martingale
part.

Approximation of quadratic variation
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
BrownianMotion, localtime
An estimator
based onquadratic
variation
Application tovolatilitymodeling

For fixed n ∈ N, we consider the time grid 0, n1 , n2 , . . . 1. For
any processes M, M̄ we set[M, M̄]n1 =nX(Mk/n − M(k−1)/n )(M̄k/n − M̄(k−1)/n ).

k=1

This is an estimator of the quadratic covariation of M, M̄. We
also writenX[M]n1 =(Mk/n − M(k−1)/n )2 ,k=1

and this is a classic estimator of the quadratic variation.

Statisticalestimation ofthe Oscillating
BrownianMotion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
BrownianMotion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modeling

We setZ t

Z t
1(Ys ≥ 0)σ(Ys )dWs = σ+ξt =01(Ys ≥ 0)dWs0

This is a martingale with quadratic variation
Z t
Z t
22
hξit =σ(Ys ) 1(Ys ≥ 0)ds = σ+1(Ys ≥ 0)ds00

From classic results on martingales (Discretization of processes,
Jacod, Protter, 2012), we have the following convergences forn → ∞:Z 1n p21(Ys ≥ 0)ds(LLN) [ξ]1 −→ hξi1 = σ+0Z 1√sl √n(CLT )n ([ξ]1 − hξi1 ) −→ 2σ+ 1(Ys ≥ 0)dBs0

where B is an independent BM.

Estimation on Y +Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

With our definition of ξ,
1Yt+ = ξt + LY
2 t
We do not observe ξ but Y + . We have

Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling[Y + ]nt = [ξ]nt −[LY ]n1+ [Y + , LY ]n1 .
4

LYt is increasing and does not contribute to the limit
Z 1+ n p
22 +[Y ]1 −→ σ+Q11(Ys ≥ 0)ds = σ+0

where we set Q1+ = Leb{s ∈ [0, 1] : Ys ≥ 0}. For 0 < u < 1P(Q1+ ∈ du) =1
1σ+ /σ−pdu.
π u(1 − u) 1 − (1 − (σ+ /σ− )2 )uEstimation of occupation time
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtimeAn estimator
based onquadratic
variation
Application tovolatility
modeling

Riemann sums
Q̄1n (Y , +) =nX1(Yk/n ≥ 0)

nk=1

converge a.s. to the Lebesgue integral
Z 1a.s.
n1(Ys ≥ 0)ds = Q1+ .
Q̄1 (Y , +) −−→
0n , the estimator for σ , asWe define now σ̂++n 2(σ̂+) =[Y + ]n1Q̄1n (Y , +)n , the estimator for σ . We have
We can define analogously σ̂−−pnnσ̂ n = (σ̂+, σ̂−)−→ (σ+ , σ− )Rate of convergence?
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Our estimator for σ+ is

Paolo Pigato
Oscillating
Brownian
Motion, local
timeAn estimator
based onquadratic
variation
Application tovolatility
modelingn 2) =(σ̂+[Y + ]n1Q̄1n (Y , +)Problem: in CLT, convergence in law! Cannot divide by arandom sample size.

yStable convergence in lawStable convergence (Rényi)
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modelingn ∈ N, Zn r.v defined on the same probability space (Ω, F, P)
Zn converges stably in law to Z if:
EYf (Zn ) → ẼYf (Z )(Z is a random variable defined on an extension, (Ω̃, F̃, P̃r))
for all bounded continuous functions f and all bounded randomvariables Y on (Ω, F ).
Stable convergence in law implies convergence in law
if Zn and Z , Yn and Y are r. v. s.t.
Zn → Z , stable in law

Yn → Y , in probabilitythen(Yn , Zn ) → (Y , Z )stable in law

Central Limit Theorem
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modeling

Estimator:n 2(σ̂+) =where[Y + ]nt = [ξ]nt −[Y + ]n1Q̄1n (Y , +)[LY ]n1+ [Y + , LY ]n1 .
4

Martingale part: stable in law convergenceZ t√sl √n(CLT )n ([ξ]1 − hξi1 ) −→ 2σ+ 1(Ys ≥ 0)dBs0

Local time?
Occupation time?

Local time part
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modeling

We prove the following convergence:√ √2 2[LY ]n1σ+ σ−p+ Y n+ [Y , L ]1 −→− √LYn −1
43 π σ+ + σ−
adapting techniques from Rates of convergence to the local
time of a diffusion, Jacod, 1998, and using convergence resultsfor discretization of martingales (see for example Limit theorem
for stochastic processes, Jacod, Shiryaev)

Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling

Summing up
√n [Y + ]n1 − hξi1
√ Z 1
2 2σ+ σ−sl √2−→ 2σ+ 1(Ys > 0)d B̄s − √LY
1 .
3 π σ+ + σ−0

Paolo Pigato

Recall the estimator
Oscillating
BrownianMotion, local
timeAn estimator
based onquadratic
variation
Application tovolatility
modelingn 2) =(σ̂+whereQ̄1n (Y , +) =[Y + ]n1Q̄1n (Y , +)nX1(Yk/n ≥ 0)
k=1

n

is an approximation of the occupation time
Q1+ = Leb(s ∈ [0, 1] : Ys ≥ 0)

Speed of convergence for occupation time
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modeling

For SDEs with smooth coefficients, the speed of convergence ofthe occupation time is n3/4− (Ngo, Ogawa), but there are noresults for discontinuous coefficients.
We prove that for Y OBM with Y0 = 0, the following
convergence holds:
 p√→0n Q̄1n (Y , +) − Q1+ −again with techniques involving local time and martingales.

Main theorem
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtimeAn estimator
based onquadratic
variation
Application tovolatility
modeling

The following convergence holds √2σ2 R1
 n 2
+21(Y>0)dB̄+ss√(σ̂+ ) − σ+0sl
Q1n−→  √2σ2 Rn )2 − σ 21(σ̂−−−1(Y < 0)d B̄
−ss
1−Q1+ 0!
√
1σ− σ+2 2Q1+
√L1 (Y ),13 π σ+ + σ−
1−Q1+

where B̄ is a BM independent of Y .
Occupation time ⇔ actual sample size

Main theorem
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtime
An estimator
based onquadraticvariation
Application tovolatility
modeling

We can rewrite such convergence as follows: √
√
22σ+1 √ ξ 1−Λ
8 n 2√√N1 − 3 π r +12√(σ̂+ ) − σ+(1−Λ)+Λr 2l  √Λ n→−√n222
2σ(σ̂− ) − σ−
ξ
Λ
18−
√
N2 − √ 1/r +1 √
21−Λ

3 π

Λ+(1−Λ)/r

where r = σ+ /σ− , ξ, N1 , N2 , Λ are mutually independent,ξ ∼ exp(1), N1 , N2 ∼ N(0, 1) and Λ follows the modified
arcsine law with density given by:pΛ (τ ) =1rπτ 1/2 (1 − τ )1/2 1 − (1 − r 2 )τ.

Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

An asymptotic bias is present in σ̂ n . This bias has the same√order (∼ 1/ n) as the ‘natural fluctuations” of the estimator.
n has a probability greater
Since the local time is positive, σ̂+n.
than 1/2 to be smaller than σ+ , and the same holds for σ̂−A modified estimator
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
BrownianMotion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

We define now a different estimator for σ+ :ss
[Y + , Y ]n1[Y − , Y ]n1nnm+ =,m− =nQ̄1 (Y , +)Q̄1n (Y , −)The following convergence holds for n → ∞: √2σ2 R
1
+n221(Y>0)dB̄+ss√(m+ ) − σ+0slQ1n−→  √2σ2 Rn )2 − σ 21(m−−−1(Y < 0)d B̄
1−Q1+

where B̄ is a BM independent of Y .

0ss

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

We can rewrite such convergence as follows:
 √ 2

2σ+
n22√
N
√
1(m+ ) − σ+ln→−  √2σΛ2n )2 − σ 2(m−−−√N1−Λ

2N1 , N2 , Λ are mutually independent, N1 , N2 ∼ N(0, 1) and Λ
follows the modified arcsine law.

Comparison between the estimators
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, localtimeAn estimator
based onquadratic
variation
Application tovolatility
modeling√n )2 − σ 2 ) (dashed) andn((σ̂++√n )2 − σ 2 ) (solid)n((m++Comparison between the estimators and thetheoretical limit distribution
Statisticalestimation of
the Oscillating
BrownianMotion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling√n )2 − σ 2 ) (dashed),n((σ̂++theoretical limit (red)√n )2 − σ 2 ) (solid) and then((m++Application to volatility modeling
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

In the Black & Scholes Model, the detrended log-price follows
dXt = σdWt
with σ positive constant, W Brownian Motion. One possible
generalization of this model is to let σ depend on the pricevariable X (local volatility model). The oscillating Brownian
motion can be seen as an example of such models, withσ(x) =+ for x ≥ 0
σ− for x < 0n σ

Simplest way to account ofLeverage effect (volatility negatively correlated with thevalue of the stock)
Volatility clusteringLiterature on regime switching models
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatilitymodeling
Paolo Pigato
Oscillating
BrownianMotion, localtime
An estimator
based onquadratic
variation
Application tovolatility
modeling

Large literature on threshold models: thresholdautoregressive models (TAR) and especially self exciting
TAR (SETAR), H. Tong, . . .
Self exciting threshold interest rates models, M. Decamps,
M. Goovaerts, and W. Schoutens.
Relation between the SET-Vasicek model and the OBM?
Filling the gaps, A. Lipton and A. Sepp.
Tiled volatility models are considered in connection withoption pricing and implied volatility
On a continuous time stock price model with regime
switching, delay, and threshold, P. P. Mota and M. L.
Esquivel.

Maximum likelihood estimation of the threshold
Statisticalestimation ofthe Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

Given an empirical time series X = (Xt )t , we do not only
estimate the parameters of the OBM, but also the threshold,using a MLE method.
For a fixed threshold r , we consider the time series X − r andestimate on it σ̂+ , σ̂− , using our estimator. We then computethe log-likelihoodXΛ(r ) =log p(Xi , Xi+1 , σ+ , σ− , r ),iand chose as threshold the level rˆ maximizing Λ.

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Log-likelihood Λ(r ) for Procter & Gamble
3580

Paolo PigatoAn estimator
based onquadratic
variation

Log−likelihood / PG

3570

Oscillating
Brownian
Motion, local
time

3560

3550

Application tovolatility
modeling
3540

5060

Threshold70Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Price and threshold for Procter & Gamble

70

Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

PG

Paolo Pigato

60

50

2005

2006

2007

2008

year

2009

2010

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Log-likelihood Λ(r ) for CA Technologies Inc

3100

Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

Log−likelihood / CA

Paolo Pigato

30503000

152025ThresholdStatisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Price and threshold for CA Technologies Inc
30

Paolo Pigato
25

An estimator
based onquadratic
variationCA

Oscillating
Brownian
Motion, local
time

20

Application tovolatility
modeling

15

2005

2006

2007

2008

year

2009

2010

Comparison with Mota Esquivel
Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Table: Estimated parameters

Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modelingStockP&G
McDonalds
CA IncMicrosoft
Citigroup

r62.24
52.4
25.07
21.8
40.7

OBM
σ−
0.014
0.013
0.025
0.034
0.075

σ+
0.012
0.018
0.013
0.017
0.011

r61.9
54.6
22.16
22.8
43.1

RS
σ−
0.013
0.014
0.033
0.034
0.076

σ+
0.013
0.016
0.015
0.016
0.011

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Log-likelihood Λ(r ) for S&P 500
9100

Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatilitymodeling

Log−likelihood / S&P 500

Paolo Pigato
9000

8900

8800

1000

1400

Threshold

1800

2200

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling

Price and threshold for S&P 500

2000

Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation

S&P 500

Paolo Pigato

1500

1000

Application tovolatility
modeling
2006

2008

2010

2012

2014

2016

year

The algorithm detects the 2009 crisis!

Statisticalestimation of
the Oscillating
Brownian
Motion andapplication tovolatility
modeling
Paolo Pigato
Oscillating
Brownian
Motion, local
time
An estimator
based onquadratic
variation
Application tovolatility
modeling

Thanks!
paolo.pigato@inria.fr
Post-doc at INRIA Nancy, equipe TOSCA.